Question

Calculate the line integral of F=−5i−3j along the line segment from (-2,5) to (-1,5).

a. 10
b. -5
c. -10
d. 5

Answer 1

The line integral of the vector field F = -5i - 3j along the line segment from (-2, 5) to (-1, 5) is calculated using the line integral formula.

Step-by-step explanation:

The student is asked to calculate the line integral of the vector field F = -5i - 3j along the line segment from (-2, 5) to (-1, 5). To do this, we can use the formula for line integrals in a vector field, which is:

\(\int_C \mathbf{F} \cdot d\mathbf{r} \)

Here, \(\mathbf{r}\) traces the path from the initial to the final point, and \(d\mathbf{r}\) is the differential element along the path. Since the movement is only in the x-direction, the differential element can be expressed as \(dx \mathbf{i}\). Substituting the vector field and the differential element into the formula gives:

\(\int_{-2}^{-1} -5 \, dx\)

Integrating, we find:

\([-5x]_{-2}^{-1} = -5(-1) - (-5)(-2)\)

\(= 5 - 10 = -5\)

Therefore, the line integral of the given vector field along the specified path is -5, which corresponds to option b.

Answer 2

b. -5

Step-by-step explanation:

To calculate the line integral of the vector field, F = -5i − 3j, along the line segment from (−2, 5) to (−1, 5), we can use:

`\displaystyle \Longrightarrow \int_C {\vec F \cdot d\vec r} \,`

Where,

• F = −5i - 3j
• dr = idx + jdy

• (-2, -1) ≤ (x, y) ≤ (-1, 5)

Plug in these values to solve:

`\displaystyle \Longrightarrow \int\limits^((-1,5))_((-2,5)) {(-5 \hat i-3\hat j) \cdot (\hat idx+\hat j dy)} \\\\\\\\\Longrightarrow \int\limits^((-1,5))_((-2,5)) {[(-5 \hat i)( \hat idx)+(-3\hat j)(\hat j dy)]} \\\\\\\\\Longrightarrow -\left[\int\limits^(-1)_(-2) {5} \, dx+ \int\limits^(5)_(5) {3} \, dy \right]\\\\\\\\\Longrightarrow -\left[[5x]\limits^(-1)_(-2)+ 0 \right]\\\\`

`\displaystyle \Longrightarrow -[5(-1)-5(-2)]\\\\\\\\\Longrightarrow -[-5+10]\\\\\\\\\Longrightarrow -[5]\\\\\\\\\therefore \int_C {\vec F \cdot d\vec r}=\boxed{-5}`

Thus, the correct option is b. -5.

`\hrulefill`

Alternatively, we can use the Fundamental Theorem for line integrals.

We are given:

• F = −5i - 3j

• (-2, -1) ≤ (x, y) ≤ (-1, 5)

Our force is conservative (∵ P_x = Q_y). So we can use the potential function to find the value of our line integral. Our potential function is:

`\Longrightarrow f(x,y)= -5x-5y`

Using the following formula we can solve this problem:

`\displaystyle \Longrightarrow \int_C {\vec F \cdot d\vec r} \, = f(\vec r(b))-f(\vec r(a))\\\\\\\\\Longrightarrow (-5(-1)-3(5))-(-5(-2)-3(5))\\\\\\\\\Longrightarrow (5-15)-(10-15)\\\\\\\\\Longrightarrow (-10)-(-5)\\\\\\\\\therefore = \boxed{-5}`

Thus, we get the same answer of -5. Please note we can only do this method when the force vector is conservative.